Geometry Unit 2 Study Guide: Lines & Angle Relationships

Practice Problem Drills

Day 1 & 2: Angle Identification & Calculation

Key Concepts for this Section

When lines $l$ and $m$ are parallel:

Congruent Angles (equal): Corresponding, Alternate Interior, Alternate Exterior, Vertical.
Supplementary Angles (add to $180^\circ$): Consecutive Interior, Linear Pair.

1. In the diagram, given the $115^\circ$ angle, find $m\angle b$ and $m\angle d$.

2. Find $m\angle e$ and $m\angle h$.

3. What is the relationship between the $115^\circ$ angle and angle 'e'?

4. What is the relationship between angle 'c' and angle 'f'?

5. What is the relationship between angle 'b' and angle 'g'?

6. If two parallel lines are cut by a transversal, are consecutive interior angles congruent or supplementary?

7. Name a pair of angles that are vertical.

8. If $m\angle c = 65^\circ$, what is $m\angle f$? Explain your reasoning.

9. Name all the angles that are supplementary to $\angle g$.

10. Name all the angles that are congruent to the $115^\circ$ angle.

Day 3: Proving Lines Parallel

Key Concept: The Converses

To prove lines are parallel, you must show that one of the angle pair relationships is true. For example, if you can show that a pair of Alternate Interior Angles are congruent, then the lines are parallel by the Converse of the Alternate Interior Angles Theorem.

11. If $m\angle 4 = 120^\circ$ and $m\angle 5 = 120^\circ$, are the lines parallel? Explain which theorem or postulate proves it.

12. If $m\angle 3 = 85^\circ$ and $m\angle 5 = 95^\circ$, are the lines parallel? Explain your reasoning.

13. If $m\angle 2 = 75^\circ$ and $m\angle 6 = 75^\circ$, what postulate proves the lines are parallel?

14. If $m\angle 1 = 105^\circ$ and $m\angle 8 = 105^\circ$, what theorem proves the lines are parallel?

15. If $m\angle 4 = (3x+10)^\circ$ and $m\angle 6 = (4x-5)^\circ$, what value of $x$ makes the lines parallel?

16. If $m\angle 3 = 50^\circ$ and $m\angle 6 = 130^\circ$, are the lines parallel? Why or why not?

17. Find the value of $x$ that makes the lines parallel if $m\angle 1 = (2x + 15)^\circ$ and $m\angle 5 = (3x - 5)^\circ$.

18. If $m\angle 2 = 110^\circ$ and $m\angle 8 = 70^\circ$, are the lines parallel? Why or why not?

19. If $m\angle 4 = m\angle 8$, does this guarantee the lines are parallel? Explain why or why not.

20. Find the value of $x$ that makes the lines parallel if $m\angle 3 = (10x - 17)^\circ$ and $m\angle 5 = (8x + 1)^\circ$.

Day 4: Solving Problems Using Angle Relationships

Key Concept: Setting up Equations

If angles are Congruent, set them equal: Expression A = Expression B.
If angles are Supplementary, add them to 180: Expression A + Expression B = 180.

21. Two alternate exterior angles measure $(5x - 50)^\circ$ and $(2x + 10)^\circ$. Find $x$.

22. Two corresponding angles measure $(x + 15)^\circ$ and $(2x - 5)^\circ$. Find the measure of each angle.

23. Two consecutive interior angles measure $(3y + 5)^\circ$ and $(2y)^\circ$. Find $y$.

24. Angle 1 and Angle 2 are a linear pair. If $m\angle 1 = 7x$ and $m\angle 2 = 5x$, find the measure of both angles.

25. If $m\angle 4 = (4z + 12)^\circ$ and $m\angle 5 = (z + 8)^\circ$. Find $m\angle 2$.

26. If $m\angle 2 = (6x - 20)^\circ$ and $m\angle 7 = (4x)^\circ$. Find $x$.

27. If $m\angle 3 = (2x + 23)^\circ$ and $m\angle 5 = (3x - 3)^\circ$, find the measure of $\angle 3$.

28. If $m\angle 1 = (8x - 12)^\circ$ and $m\angle 7 = (7x + 2)^\circ$, find the measure of $\angle 1$.

29. If $m\angle 4 = (5x - 10)^\circ$ and $m\angle 8 = (3x + 20)^\circ$, find $x$. (Hint: What is the relationship between $\angle 4$ and $\angle 6$?)

30. If $m\angle 2 = 4y - 14$ and $m\angle 8 = 3y+1$, find $m\angle 7$.

21. $5x - 50 = 2x + 10 \rightarrow 3x = 60 \rightarrow x = 20$.

22. $x + 15 = 2x - 5 \rightarrow x = 20$. Angles: $(20)+15 = 35^\circ$ and $2(20)-5 = 35^\circ$.

23. $3y + 5 + 2y = 180 \rightarrow 5y = 175 \rightarrow y = 35$.

24. $7x + 5x = 180 \rightarrow 12x = 180 \rightarrow x = 15$. Angles: $7(15) = 105^\circ$ and $5(15) = 75^\circ$.

25. $\angle 4$ and $\angle 5$ are alternate interior. $4z + 12 = z + 8 \rightarrow 3z = -4 \rightarrow z = -\frac{4}{3}$. $m\angle 4 = 4(-\frac{4}{3})+12 = -\frac{16}{3} + \frac{36}{3} = \frac{20}{3}$. $\angle 2$ and $\angle 4$ are vertical, so $m\angle 2 = \frac{20}{3}^\circ$.

26. $\angle 2$ and $\angle 7$ are consecutive exterior, so they are supplementary. $(6x-20)+(4x)=180 \rightarrow 10x - 20 = 180 \rightarrow 10x=200 \rightarrow x=20$.

27. Consecutive interior are supplementary. $(2x+23)+(3x-3)=180 \rightarrow 5x+20=180 \rightarrow 5x=160 \rightarrow x=32$. $m\angle 3 = 2(32)+23 = 64+23 = 87^\circ$.

28. Vertical angles are congruent. $m\angle 1 = m\angle 3$. Alternate interior angles are congruent. $m\angle 3 = m\angle 6$. Supplementary angles. $m\angle 6 + m\angle 7 = 180$. So, $m\angle 1 + m\angle 7 = 180$. $(8x-12)+(7x+2)=180 \rightarrow 15x-10=180 \rightarrow 15x=190 \rightarrow x \approx 12.67$. $m\angle 1 = 8(12.67) - 12 \approx 101.33-12 = 89.33^\circ$.

29. $\angle 4$ and $\angle 6$ are consecutive interior. $\angle 6$ and $\angle 8$ are corresponding, so $m\angle 6 = m\angle 8$. So, $m\angle 4 + m\angle 8 = 180$. $(5x-10)+(3x+20)=180 \rightarrow 8x+10=180 \rightarrow 8x=170 \rightarrow x = 21.25$.

30. $\angle 2$ and $\angle 8$ are consecutive exterior. $m\angle 2 + m\angle 8 = 180$. $(4y-14)+(3y+1)=180 \rightarrow 7y-13=180 \rightarrow 7y=193 \rightarrow y \approx 27.57$. $m\angle 8 = 3(27.57)+1 \approx 83.71^\circ$. $\angle 7$ and $\angle 8$ are a linear pair. $m\angle 7 = 180 - 83.71 = 96.29^\circ$.

Day 5: Midpoint & Distance Formula

Formulas for this Section

Midpoint Formula

$$ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$

Distance Formula

$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

31. Find the midpoint and distance: $A(-2, 3)$ and $B(4, -5)$.

32. Find the midpoint and distance: $C(5, 2)$ and $D(1, 10)$.

33. Find the midpoint and distance: $E(0, -4)$ and $F(3, 0)$.

34. Find the midpoint and distance: $G(-3, -1)$ and $H(-7, -5)$.

35. A line segment has an endpoint at $(2, 8)$ and a midpoint at $(5, 2)$. Find the other endpoint.

36. Find the perimeter of a triangle with vertices at $P(1,5), Q(5,2),$ and $R(1,2)$.

37. Find the midpoint and distance: $J(\frac{1}{2}, 3)$ and $K(\frac{5}{2}, -1)$.

38. What is the length of a line segment with endpoints $(-1, -1)$ and $(5, 7)$?

39. Segment RS has an endpoint $R$ at $(-9, 8)$ and a midpoint $M$ at $(-1, 5)$. Find the coordinates of endpoint $S$.

40. A triangle has vertices at $A(1, 2), B(5, 5),$ and $C(5, -1)$. Find the length of each side.